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Question
Find the second order derivatives of the following function ex sin 5x ?
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Solution
We have,
\[y = e^x \sin\left( 5x \right)\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = e^x \sin 5x + e^x \cos 5x \times 5\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = e^x \sin 5x + e^x \cos 5x \times 5 + 5 e^x ( - \sin5x \times 5) + 5 e^x \cos 5x \]
\[ = - 24 e^x \sin 5x + 10 e^x \cos 5x\]
\[ = 2 e^x \left( 5 \cos 5x - 12 \sin 5x \right)\]
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